Relaxation in an L∞-optimization problem

2003 ◽  
Vol 133 (3) ◽  
pp. 599-615 ◽  
Author(s):  
Hitoshi Ishii ◽  
Paola Loreti
Keyword(s):  
Optimization Problem ◽  
Dirichlet Boundary ◽  
Continuous Functions ◽  
Maximization Problem ◽  
Bounded Subset ◽  
Lipschitz Continuous Functions ◽  
Lipschitz Continuous ◽  
Image Position ◽  
Open Bounded Subset ◽  
Asymptotic Problem

Let Ω be an open bounded subset of Rn and f a continuous function on Ω̄ satisfying f(x) > 0 for all x ∈ Ω̄. We consider the maximization problem for the integral ∫Ωf(x)u(x)dx over all Lipschitz continuous functions u subject to the Dirichlet boundary condition u = 0 on ∂Ω and to the gradient constraint of the form H(Du(x)) ≤ 1, and prove that the supremum is ‘achieved’ by the viscosity solution of Ĥ(Du(x)) = 1 in Ω and u = 0 on ∂Ω, where Ĥ denotes the convex envelope of H. This result is applied to an asymptotic problem, as p → ∞, for quasi-minimizers of the integral An asymptotic problem as k → ∞ for inf is also considered, where the infimum is taken all over and the set K is given by {ξ | H(ξ) ≤ 1}.

scholarly journals The Rate of Convergence of Lupasq-Analogue of the Bernstein Operators

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Heping Wang ◽  
Yanbo Zhang
Keyword(s):  
Rate Of Convergence ◽  
Modulus Of Continuity ◽  
Continuous Functions ◽  
Bernstein Operators ◽  
Lipschitz Continuous Functions ◽  
Lipschitz Continuous

We discuss the rate of convergence of the Lupasq-analogues of the Bernstein operatorsRn,q(f;x)which were given by Lupas in 1987. We obtain the estimates for the rate of convergence ofRn,q(f)by the modulus of continuity off, and show that the estimates are sharp in the sense of order for Lipschitz continuous functions.

scholarly journals A new metric between distributions of point processes

2008 ◽  
Vol 40 (03) ◽  
pp. 651-672 ◽  
Author(s):  
Dominic Schuhmacher ◽  
Aihua Xia
Keyword(s):  
Point Processes ◽  
Relative Difference ◽  
Continuous Functions ◽  
Upper And Lower Bounds ◽  
Point Pattern ◽  
Multiple Point ◽  
Finite Point ◽  
Lipschitz Continuous Functions ◽  
Lipschitz Continuous ◽  
Comprehensive Collection

Most metrics between finite point measures currently used in the literature have the flaw that they do not treat differing total masses in an adequate manner for applications. This paper introduces a new metric d̅ 1 that combines positional differences of points under a closest match with the relative difference in total mass in a way that fixes this flaw. A comprehensive collection of theoretical results about d̅ 1 and its induced Wasserstein metric d̅ 2 for point process distributions are given, including examples of useful d̅ 1-Lipschitz continuous functions, d̅ 2 upper bounds for the Poisson process approximation, and d̅ 2 upper and lower bounds between distributions of point processes of independent and identically distributed points. Furthermore, we present a statistical test for multiple point pattern data that demonstrates the potential of d̅ 1 in applications.

EXISTENCE OF A WEAK SOLUTION FOR A CLASS OF FRACTIONAL LAPLACIAN EQUATIONS

2016 ◽  
Vol 102 (3) ◽  
pp. 392-404
Author(s):  
V. RAGHAVENDRA ◽  
RASMITA KAR
Keyword(s):  
Weak Solution ◽  
Fractional Laplacian ◽  
Nonlocal Problem ◽  
Bounded Subset ◽  
Lipschitz Boundary ◽  
Integrodifferential Operator ◽  
Image Position ◽  
Laplacian Equations ◽  
Open Bounded Subset ◽  
Fractional Type

We study the existence of a weak solution of a nonlocal problem$$\begin{eqnarray}\displaystyle & \displaystyle -{\mathcal{L}}_{K}u-\unicode[STIX]{x1D707}ug_{1}+h(u)g_{2}=f\quad \text{in }\unicode[STIX]{x1D6FA}, & \displaystyle \nonumber\\ \displaystyle & \displaystyle u=0\quad \text{in }\mathbb{R}^{n}\setminus \unicode[STIX]{x1D6FA}, & \displaystyle \nonumber\end{eqnarray}$$where${\mathcal{L}}_{k}$is a general nonlocal integrodifferential operator of fractional type,$\unicode[STIX]{x1D707}$is a real parameter and$\unicode[STIX]{x1D6FA}$is an open bounded subset of$\mathbb{R}^{n}$($n>2s$, where$s\in (0,1)$is fixed) with Lipschitz boundary$\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$. Here$f,g_{1},g_{2}:\unicode[STIX]{x1D6FA}\rightarrow \mathbb{R}$and$h:\mathbb{R}\rightarrow \mathbb{R}$are functions satisfying suitable hypotheses.

scholarly journals Imbeddings of Brezis-Wainger type. The case of missing derivatives

2001 ◽  
Vol 131 (3) ◽  
pp. 667-700
Author(s):  
M. Krbec ◽  
Hans-Jürgen Schmeisser
Keyword(s):  
Continuous Functions ◽  
Lipschitz Continuous Functions ◽  
Lipschitz Continuous ◽  
Mixed Derivatives

We prove limiting imbeddings of spaces with dominating mixed derivatives into the spaces of almost Lipschitz continuous functions.

scholarly journals Clarke critical values of subanalytic Lipschitz continuous functions

2005 ◽  
Vol 87 ◽  
pp. 13-25 ◽  
Author(s):  
Jérôme Bolte ◽  
Aris Daniilidis ◽  
Adrian Lewis ◽  
Masahiro Shiota
Keyword(s):  
Continuous Functions ◽  
Critical Values ◽  
Lipschitz Continuous Functions ◽  
Lipschitz Continuous

scholarly journals Continuity and differentiability properties of the Nemitskii operator in Hölder spaces

1988 ◽  
Vol 30 (1) ◽  
pp. 59-65 ◽  
Author(s):  
Rita Nugari
Keyword(s):  
Banach Space ◽  
Euclidean Space ◽  
Dimensional Euclidean Space ◽  
Bounded Subset ◽  
Holder Space ◽  
Usual Norm ◽  
Image Position ◽  
Holder Spaces ◽  
Open Bounded Subset ◽  
Differentiability Properties

Let ℝn be the n-dimensional Euclidean space with the usual norm denoted by |·| In what follows 蒆 will denote an open bounded subset of ℝn, and its closure.For α ∊(0,1], is the space of all functions such that: is called the Holder space with exponent a and is a Banach space when endowed with the norm:where ‖u‖∞ is, as usual, defined by:

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